Define the segments of a stepped shaft, its bearings and mounted elements (gears, pulleys, couplings) and the tool computes the forces in the horizontal and vertical planes, the moment and torque diagrams, the stresses at every shoulder, keyseat and groove with their stress concentrations, and the fatigue safety factor by Goodman, Soderberg, Gerber and ASME-elliptic.
Because as it rotates, every surface point goes from tension to compression each turn: the bending stress is fully reversed.
Modified Goodman is the most common in design; Soderberg is more conservative and Gerber or ASME-elliptic fit test data better.
A shaft that turns with gears or pulleys carries bending and torsion. Even with constant loads, as it rotates every fiber goes from tension to compression once per revolution: the bending stress is fully reversed. After millions of cycles the shaft can break by fatigue at stresses well below yield. This guide summarizes Shigley's method for checking a shaft and applies it to the example the tool opens with.
Gear forces (tangential and radial) and pulley forces act in two planes. Bearing reactions and the moment diagrams in each plane are computed and combined: M = √(Mxy² + Mxz²). The torque T runs between the elements that put it in and those that take it out.
In a round section of diameter d, the bending stress is σ = 32M/(πd³) and the torsional shear τ = 16T/(πd³). At shoulders, keyways and grooves the stress concentrates: Kt comes from charts (Peterson) and, in fatigue, Kf = 1 + q(Kt − 1) is used, where q is the material's notch sensitivity.
For steels, Se′ ≈ 0.5·Sut (up to 1400 MPa). The value for the real part is corrected with the Marin factors:
Se = ka·kb·kc·kd·ke·Se′
ka for surface finish, kb for size, kc for load type, kd for temperature and ke for reliability.
With the alternating and mean von Mises stresses σ′a and σ′m, the DE-Goodman criterion gives the safety factor:
1/n = σ′a/Se + σ′m/Sut
Soderberg (more conservative, uses Sy), Gerber and ASME elliptic (less conservative) are alternatives. First-cycle yielding is also checked: ny = Sy/σ′max.
In the example the tool opens with, a machined countershaft of cold-drawn AISI 1050 steel (Sut = 690 MPa, Sy = 580 MPa), the critical point is a keyway at x = 200 mm, with d = 42 mm, M = 283.5 N·m and a steady T = 200 N·m.
These are the values in the tool's critical points table, which repeats this calculation at every shoulder, keyway and groove and flags the worst one.
A strong shaft can still be too flexible: if it deflects too much, gears mesh poorly and bearings misalign. That is why the deflection at the gears, the slope at the bearings and the critical speed, which must be well above the running speed, are checked.
R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design (McGraw-Hill), the fatigue and shaft chapters; R. L. Mott, Machine Elements in Mechanical Design (Pearson). Simuveliq is not affiliated with these authors or publishers.
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